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Aggregate excess demand subtracts total endowment from the sum of consumers' demand correspondences. A market-clearing price satisfies $z(p)=0$. Under Walras law, one equation is redundant after price normalization.

If zero is a regular value of the joint smooth aggregate excess demand $Z(p,t)$, almost every parameter gives a regular economy. The regular level set theorem makes the zero set a manifold. Its projection to parameter space fails to have full rank exactly where the derivative with respect to prices is singular; Sard theorem makes these exceptional parameter values a null set.

Normalize the second price to one and write $p>0$ for the first price and $t>0$ for the varying endowment. Let $Z(p,t)$ be first-good aggregate excess demand. Under Walras law, the second excess-demand coordinate equals $-pZ(p,t)$, so one market-clearing equation suffices. A regular exchange economy at fixed $t$ has $Z_p(p,t)\ne 0$ at every equilibrium.

A sufficient joint condition is that zero be a regular value of the smooth map $Z:(0,\infty {)}^2\to \mathbb{R}$:To prove the almost-everywhere conclusion, the regular level set theorem makes $M=Z^{-1}(0)$ a smooth one-dimensional manifold. Project $M$ to the parameter axis by $\pi (p,t)=t$. Its tangent space consists of vectors $(h,k)$ with $Z_{p}h+Z_{t}k=0$. If $Z_p\ne 0$, every $k$ is possible, so the derivative of $\pi$ is surjective. If $Z_p=0$, the joint condition gives $Z_t\ne 0$, forcing $k=0$; the point is critical for $\pi$.

Thus the parameters with a singular equilibrium are exactly the critical values of this projection. Sard theorem makes that set a null set. Therefore almost every parameter gives a regular economy. No claim of existence at every parameter is needed for this sufficient regularity criterion.

Write the first endowment as $(a,t)$. The budget constraint at normalized prices is $px+y\le pa+t$. Because the quasilinear utility is increasing in $y$, the budget binds, and its maximization reduces to maximizing $\log x-px+pa+t$ over $x>0$. Its derivative is $1/x-p$ and its second derivative is $-1/x^2<0$, giving the unique demand correspondenceThe first-good demand has no wealth effect. Since the second consumer's endowment and preference relation are fixed, $Z_t=0$ everywhere. Thus the joint condition of part (a) becomes $Z_p\ne 0$ at equilibrium; it is not automatically satisfied.

Here is a complete smooth counterexample, with the second consumer's consumption set specified explicitly. The question does not impose the first consumer's consumption set on the second consumer. Take endowments $(a,t)=(1/10,t)$ and $(b,c)=(2,1)$. Give the second consumer consumption set $X_2=(-\infty ,A)\times \mathbb{R}$, where $A=43/20$, and put $B=31/20$. Its utility function isThis defines complete, transitive, continuous preference relations. It is in fact smooth on $X_2$: writing $r=A-x>0$ and $s=y-B>0$, the positive branch is $\exp (-r^2/(4s)-r/2-s/4)$; at $s=0$ all derivatives tend to zero locally uniformly where $r>0$. The endowment belongs to $X_2$. Allowing negative holdings is explicit here, just as the first consumer's second coordinate is unrestricted in the printed question.

We now derive, rather than posit, the second consumer's demand correspondence. For $v>0$, the upper contour $U_2\ge e^{-v}$ is described by $(r+s{)}^2\le 4vs$, with $r,s>0$. At a fixed $s$, its cheapest first coordinate uses $r=2\sqrt{vs}-s$. Minimizing total expenditure over $s$ therefore means minimizingThe unique minimum occurs at $\sqrt{s}=p\sqrt{v}/(1+p)$, with $r=vp(p+2)/(1+p{)}^2>0$. The expenditure necessary to reach this contour is consequentlySet $\delta =A-b=3/20$ and $\epsilon =B-c=11/20$. Wealth is $pb+c$, so the best affordable contour hasAll affordable positive-utility bundles have $v\ge v_{\ast }$, and the minimum-expenditure bundle on that contour is unique and spends precisely the budget. Zero-utility bundles cannot be optimal because this positive-utility bundle is available. Substitution yields globally smooth unique demands for every $p>0$:The aggregate excess demand is nowFor every $t>0$, the unique equilibrium price is $p=1$, and $Z_p(1,t)=Z_t(1,t)=0$. At this price the allocations are $(1,t-9/10)$ and $(11/10,19/10)$, whose sum equals the aggregate endowment $(21/10,t+1)$. Second-good excess demand is $-pZ$, so it too is smooth and clears. Hence every economy in this smooth parameterized family is singular. This demonstrates why the first consumer's missing wealth effect alone gives no transversality guarantee; the stronger conclusion in part (c) comes from the second consumer's additional demand restriction.

An exchange economy is regular when the derivative of its reduced aggregate excess demand has full rank at every equilibrium. For two goods and normalized prices $(p,1)$, this means $Z_p(p)\ne 0$ at every zero of first-good excess demand. Regular equilibria are locally isolated and persist smoothly under small parameter changes.